Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C _d≡ F/(ηU) = k ₀ + k ₂ Re²+ .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K (U/a)^n-1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k ₀ is the drag coefficient for creeping flows. The proportionality constant k ₂ depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k ₂ on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.     

The calculation of drag on nano‐cylinders

The study of molecular flows at low Knudsen numbers (～0.1–0.5), over nano‐scaled objects of 20–100 nm size is becoming an important area of research. The simulation of fluid–structure interaction at nano‐scale is important for understanding the adsorption and drag resistance characteristics of nano‐devices in the fields of drug delivery, surface cleaning and protein movement. A novel formulation has been proposed that calculates localised values for both the kinetic and configurational parts of the Irving–Kirkwood stress tensor at given fixed positions within the computational domain. Macroscopic properties, such as streaming velocity, pressure and drag coefficients, are predicted by modelling the fluid–structure interaction using a moving least‐squares method. The gravitation‐driven molecular flow is examined over three different cross‐sectional shapes—i.e. diamond‐, circular‐ and square‐shaped cylinders—confined within parallel walls and has been simulated for rough and smooth surfaces. The molecular dynamics formulation has allowed, for the first time, the calculation of localised drag forces over nano‐cylinders. The computational simulation has shown that existing methods, including continuum‐based approaches, significantly underestimate drag coefficients over nano‐cylinders. The proposed molecular dynamics formulation has been verified on simulation based tests, as experimental and analytical results are unavailable at this scale. Copyright © 2016 John Wiley & Sons, Ltd.     

A modified drag model for power-law fluid-particle flow used in computational fluid dynamics simulation

A modified drag model for the power-law fluid-particle flow considering effects of rheological properties was proposed. At high particle concentrations (ε_s ≥ 0.2), based on the Ergun equation, the cross-sectional shape and the tortuosity of the pore channel are considered, and the apparent flow behavior index and consistency coefficient of the power-law fluid at the surface of the particles are corrected. At low particle concentrations (ε_s < 0.2), based on the Wen-Yu drag model, the modified Reynolds number for power-law fluid and the relational expression between drag coefficient for single particle and Reynolds number that considers the effect of the flow behavior index are adopted. Numerical simulations for the power-law fluid-particle flow in the fluidized bed were carried out using the non-Newtonian drag model. The effects of rheological parameters on the drag coefficient were analyzed. The comparisons of simulation and experiment show that the modified drag model predicts reasonable void fraction under different rheological parameters, particle diameters, and liquid velocities in both low particle concentrations and high particle concentrations. The increase in flow behavior index and consistency coefficient increases the drag coefficient between the two phases and decreases the average particle concentration within the bed.     

Moving wedge and flat plate in a micropolar fluid

Similarity solutions for a moving wedge and flat plate in a micropolar fluid may be obtained when the fluid and boundary velocities are proportional to the same power-law of the downstream coordinate. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solve numerically using a finite-difference scheme known as the Keller-box method. Numerical results are given for the dimensionless velocity and microrotation profiles, as well as the skin friction coefficient for several values of the Falkner–Skan power-law parameter (m), the ratio of the boundary velocity to the free stream velocity parameter (λ) and the material parameter (K). Important features of these flow characteristics are plotted and discussed. It is found that multiple solutions exist when the boundary is moving in the opposite direction to the free stream, and the micropolar fluids display a drag reduction compared to Newtonian fluids.     

Heat transfer in boundary layer flow of a micropolar fluid past a curved surface with suction and injection

Van Dyke's singular perturbation technique has been used to study the heat transfer in the flow of a micropolar fluid past a curved surface with suction and injection. The conditions for similar solutions of the thermal boundary layer equations have been obtained. In addition to the usual “no slip” condition for velocity, the two types of boundary conditions used for microrotation are: (i) no relative spin on the boundary; (ii) the anti-symmetric part of the stress tensor vanishes at the boundary. The effect of suction or injection on velocity, microrotation, temperature, skin friction coefficient, wall couple stress coefficient, displacement and momentum thicknesses, rate of heat transfer and adiabatic wall temperature have been studied. It is observed that with the increase of injection velocity, the thickness of the boundary layer is increased and the local drag is reduced. A comparison with the results obtained for a Newtonian fluid reveals that the microelements present in the fluid reduce the velocity and frictional drag, and cool the boundary.     

Hydraulic formulae for the added masses of an impermeable sphere moving near a plane wall

Simple formulae for the components of the added-mass coefficient tensor of a sphere moving near a wall with variable velocity in an ideal fluid bounded by a solid surface are derived. The added mass is calculated numerically as a function of the dimensionless distance between the sphere and the wall for both perpendicular and parallel motions. The calculation is performed by the method of successive images. The velocity field is computed as the sum of the velocity fields of sequences of dipoles located along the axis. The obtained dependences of the added-mass tensor components are fitted by simple continuous functions with high accuracy.     

Numerical simulation of flows past periodic arrays of cylinders

We present a detailed numerical investigation of three unsteady incompressible flow problems involving periodic arrays of staggered cylinders. The first problem is a uniperiodic flow with two cylinders in each cell of periodicity. The second problem is a biperiodic flow with two cylinders in each cell, and the last problem is a uniperiodic flow with ten cylinders. Both uniperiodic flows are periodic in the direction perpendicular to the main flow direction. In all three cases, the Reynolds number based on the cylinder diameter is 100, and initially the flow field has local symmetries with respect to the axes of the cylinders parallel to the main flow direction. Later on, these symmetries break, vortex shedding is initiated, and gradually the scale of the shedding increases until a temporally periodic flow field is reached.We furnish extensive flow data, including the vorticity and stream function fields at various instants during the temporal evolution of the flow field, time histories of the drag and lift coefficients, Strouhal number, initial and mean drag coefficients, amplitude of the drag and lift coefficient oscillations, and the phase relationships between the drag and lift oscillations associated with each cylinder. Our data confirms that, at this Reynolds number, there are no stable steady-state solutions with local symmetries. Of course, one can obtain such unphysical solutions by assuming symmetry conditions along the axes of the cylinders parallel to the main flow direction and taking half of the computational domain needed normally. In such cases, the steady-state flow fields obtained would be identical to the flow fields observed at the initial stages of our computations. However, we show that such flow fields do not represent the temporally periodic flow fields even in a time-averaged sense, because, in all three cases, the initial drag coefficients are different from the mean drag coefficients. Therefore, we conclude that stability studies involving periodic arrays of cylinders should be carried out, as it is done in this work, with the true implementation of the spatial periodicity.     

Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition

This paper concerns the flow of an incompressible, viscous fluid past a porous spherical particle enclosing a solid core, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. No-slip and impenetrability boundary conditions on the solid spherical core have been used. The hydrodynamic drag force experienced by a porous spherical particle enclosing a solid core and permeability of membrane built up by solid particles with a porous shell are evaluated. It is found that the hydrodynamic drag force and dimensionless hydrodynamic permeability depends not only on the porous shell thickness, particle volume fraction γ and viscosities of porous and fluid medium, but also on the stress jump coefficient. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta–Morse’s condition). Some previous results for the hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Gravity flow of granular materials in contracting cylinders and tapered tubes

This paper presents two applications of the double-shearing theory for flow of granular materials under gravitational forces for axially symmetric flow. In the first the material is contained in a vertical circular cylinder which compresses the material by contracting radially. Exact solutions for the stress and velocity fields are derived, under the boundary conditions that the cylinder wall is either `perfectly rough' or subject to Coulomb friction. In the second problem the material flows under gravity through a tube with circular cross-section of radius that decreases slowly with depth. For this problem an approximate solution is derived that is accurate to first order in the slope of the tube wall relative to the vertical. It is also shown that an alternative theory, in which it is postulated that the principal axes of the stress and rate-of-strain tensors are coincident, leads to the prediction of unacceptable singularities in the flow field in the interior of the material.     

The slow stationary flow of incompressible micropolar fluid past a spheroid

The paper examines the slow stationary flow of incompressible micropolar fluid past a spheroid (prolate and oblate) adopting the Stokesian approximation, so that the inertial terms in the momentum equation and the bilinear terms in the balance of first stress moments are neglected. The flow over the space outside the body is analyzed and the velocity, microrotation, stress and couple stress are obtained analytically in infinite series form. The drag on the body is determined and it is observed that there is no couple exerted on the body. Numerical studies are undertaken to see the variation of the drag with respect to the geometric as well as the physical flow parameters. These have been presented in the form of figures. Micropolarity of the fluid has an augmenting effect on the drag. In an Appendix, an alternative method of determining the drag is indicated.     

Experimental and numerical analysis on the hydrodynamic behaviors of permeable microbial granules

The microbial granules are found to be porous and permeable, which leads to a different drag force coefficient from the rigid sphere granules. Discrete element method (DEM) was employed to establish geometric models of porous microbial granules for the first time in this study. And computational fluid dynamics (CFD) was applied to simulate the effects of porosity and Reynolds number on the fluid flow, shear stress, pressure and drag force based on the established geometric models. The results showed that both the Reynolds number and the porosity of microbial granules significantly affect the fluid velocity distribution inside the granules. The porosity shows less effect on maximum shear stress than Reynolds number. It s well known that drag force consists form drag and skin drag. The ratio of form drag to drag force increased, while the skin drag force ratio decreased with the increasing Reynolds number. The porosity will enhance the skin drag and weaken the form drag at the same Reynolds number. A drag force coefficient equation was established based on the simulated results in order for engineering application. The correctness of the equation was confirmed by comparing with experimental results. The results from this study may provide valuable information for operation and designing of a granule-based bioreactor in wastewater treatment.     

Electroviscous effects in purely pressure driven flow and stationary plane analysis in electroosmotic flow of power-law fluids in a slit microchannel

This paper is a comprehensive work on flow of power-law fluids in a slit microchannel. The first part of the paper deals with study of electrokinetic effects in steady, fully developed, laminar pressure driven flow of power-law fluids. The second part of the work provides analysis of stationary plane that is formed during electroosmotic flow (EOF) of power-law fluids inside a closed slit microchannel. In the entire analysis, the flow of power-law fluid is characterized by the modified Navier-Stokes equation incorporating the electric body force term. The electric double layer (EDL) potential is described by the Poisson-Boltzmann distribution under Debye Hückel approximation. In pressure driven flow, analytical expressions for velocity profiles of various power-law fluids are obtained for n = 1, 1/2, 1/3. Numerical simulation is carried out for all values of n to find induced streaming potential without any approximations for entire range of flow behavior indices. The analytical solutions are compared with numerical results. The effects of flow behavior index, zeta potential and channel dimension on velocity distribution, streaming potential, apparent viscosity, volumetric flow rate and friction coefficient are discussed. In the analysis of EOF in closed slit microchannel, the positions of stationary planes for various flow behavior indices at different EDL thicknesses are found both analytically and numerically. It is found that the electroosmotic counter pressure developed inside the cell is strongly dependent on zeta potential and weakly dependent on EDL thickness.     

Boundary layer equations and stretching sheet solutions for the modified second grade fluid

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows. The boundary layer equations are derived from the equations. Symmetries of the boundary layer equations are calculated using Lie Group theory. For a special power law index of m = −1, the principal Lie algebra extends. Using one of the symmetries, the partial differential system is transferred to an ordinary differential system. The ordinary differential equations are numerically integrated for the stretching sheet boundary conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions. The shear stress on the boundary is also calculated.     

Stream spreading in multilayer microfluidic flows of suspensions

The goal of this experimental study is to quantify the spreading of parallel streams with viscosity contrast in multilayer microfluidic flows. Three streams converge into one channel where a test fluid is sheathed between two layers of a Newtonian reference fluid. The test fluids are Newtonian fluids with viscosities ranging from 1.1 to 48.2 cP and suspensions of 10-mum-diameter PMMA particles with particle volume fractions phi = 0.16-0.30. The fluid interface locations are identified through fluorescence microscopy. The steady-state width of the center stream is strongly dependent on the viscosity ratio between the adjacent fluids and exhibits a near power-law relationship. This dependence occurs for both the Newtonian fluids and the suspensions, although the slopes differ. The high-concentration suspension (phi = 0.30) diverges from Newtonian behavior, while the low-concentration suspensions (phi = 0.16, 0.22) closely approximate that of the Newtonian fluids. The observed suspension behavior can be attributed to shear-induced particle migration.     

单侧受限管排自然对流的数值模拟

给出了具有 5,10,15,20 根管的管排在 103相似文献   

端部状态对斜拉索节段模型气动特性的影响

为了给实际工程中串列三圆柱结构的风荷载取值提供参考,通过刚性模型测压风洞试验方法,测试了12个不同间距比L/D(L为两圆柱中心之间的距离,D为圆柱的直径)下串列三圆柱的平均风压系数和平均阻力系数,并与单圆柱的平均风压系数和平均阻力系数进行了对比。研究结果表明：串列三圆柱存在临界间距,其临界间距比为3.0≤L/D≤3.5。三个圆柱的平均阻力均小于单圆柱的平均阻力;上游圆柱的平均阻力最大,下游圆柱的平均阻力次之,中游圆柱的平均阻力最小。     

两并列方柱绕流相互干扰的数值研究

为了研究并列双方柱在不同间距比时气动力特性的干扰效应,采用刚性模型测压风洞试验的方法,通过改变两方柱之间的距离,得到了不同间距比下并列双方柱的风压系数、升力系数和阻力系数。结果表明：当间距比1.2≤L/D<2.5时,并列双方柱平均风压系数的干扰效应明显,且主要表现在内侧面,平均阻力系数和脉动升力系数的干扰效应表现为减小效应;当间距比L/D≥2.5时,并列双方柱的平均风压系数、平均阻力系数和脉动升力系数的干扰效应均不明显。     

各柱做不同辐射运动时截头圆柱群的水动力分析

截头圆柱群是许多海洋浮式结构的主要组成部分。目前对这种圆柱群进行的水动力分析大多考虑将其作为一个整体,此时各柱间无相对位置变化,这并不适用于柱间有相对运动的情况。为此,采用特征展开法研究各柱做不同辐射运动时截头直立圆柱群的水动力相互作用。开展了截头圆柱群中各柱做不同幅值纵荡、横荡和垂荡运动时的波浪辐射分析,并求得流场速度势。在进行算例考核之后,进一步计算了多个圆柱同时做不同模态、不同幅值运动时截头柱群的水动力、压力分布等量,并对结果进行了分析。     

Free convection boundary layer flow of a micropolar fluid past slender bodies

The transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past vertical cylinders are investigated using the theory of micropolar fluids formulated by Eringen. The governing equations for momentum, angular momentum and energy have been solved numerically. Missing values of the velocity, angular velocity and thermal functions are tabulated for a wide range of the material parameters, transverse curvature parameter and Prandtl number of the fluid. A comparison has been made with the corresponding results for Newtonian fluids. Micropolar fluids display drag reduction and reduced surface heat transfer rate as compared with Newtonian fluids.